Meaning of Functions - AP Calculus AB

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Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

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Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Compare your answer with the correct one above

Question

Which of the following is not a function?

Answer

A function is defined when each value of x yields a single value of y (ordered pairs). The expression

$f(x)=\left\{\begin{matrix}3x+2\textup{ when } x\geq 7 & \3x-2\textup{ when } x\leq 7 \end{matrix}\right.$

is not a function because there are two values of f(x) for which a single value of x (7) creates. f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

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Question

If $h(x)=\int_{0}^{x}sin(u)du$ , find $h^{}(x)$

Answer

Taking the derivative of an integral yields the original function, but because we have a different variable in the integration limits, the variable switches

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Question

Evaluate $\int(14x^7 +\frac{x^6}{(2)}+sin(x)-csc^2(x))dx$

Answer

using integration identities: $\int{x^n dx}= \frac{x^{n+1}}{n+1} and \int{sin(x) dx}= - cos(x) and \int{csc^2(x) dx}= -cot (x)$

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Question

Evaluate the limit:

$\lim_{x \to 2} \frac{x^3-8}{x^2-4}$

Answer

First, factor both the numerator and denominator:

$\lim_{x \to 2} \frac{(x-2)(x^2+2x+4)}{(x-2)(x+2)}$

Cancel:

$\lim_{x \to 2} \frac{(x^2+2x+4)}{(x+2)}$

Plugging in 2 gives the final answer, 3.

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Question

Take the limit

$\lim_{x \to -2} \frac{x+\sqrt{x+6}}{x+2}$

Answer

First, multiply the numerator and denominator by $x-\sqrt{x+6}$ and it turns into

$\lim_{x \to -2} \frac{(x+\sqrt{x+6})}{(x+2)}\frac{(x-\sqrt{x+6})}{(x-\sqrt{x+6})}$

$\lim_{x \to -2} \frac{x^2-x-6}{(x+2)(x-\sqrt{x+6})}$

Factor the numerator and then cancel out the 'x+2'

$\lim_{x \to -2} \frac{(x+2)(x-3)}{(x+2)(x-\sqrt{x+6})}$

$\lim_{x \to -2} \frac{(x-3)}{(x-\sqrt{x+6})}$

After taking the limit, the answer is $\frac{5}{4}$

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Question

If this limit is true, then what is the value of 'a'?

$\lim_{x \to 3} \frac{x^4-a}{x^2-\sqrt{a}}=18$

Answer

Factor the numerator

$\lim_{x \to 3} \frac{(x^2-\sqrt{a})(x^2+\sqrt{a})}{x^2-\sqrt{a}}=18$

Cancel the $x^2-\sqrt{a}$ , plug in the limit and then solve for 'a'

$\lim_{x \to 3} x^2+\sqrt{a}=18$

$3^2+\sqrt{a}=18$

$a=81$

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Question

We have a line described as $y=-\frac{1}{2}x+2$ . Find the minimum distance between the origin and a point on that line.

Answer

We have the origin and a point located on the line. That point represents the minimum distance to the orgin. Apply the distance formula to these two points,

$d=\sqrt{(x-0)^2+(y-0)^2}=\sqrt{x^2+y^2}$

Plug in the line equation, take the derivative, set it equal to zero, and solve for x.

$d=\sqrt{x^2+(-\frac{1}{2}x+2)^2}=\sqrt{\frac{5}{4}x^2-2x+4}$ ${d}'=\frac{\frac{5}{2}x-2}{2\sqrt{\frac{5}{4}x^2-2x+4}}=0$

$x=\frac{4}{5}$

Use this value to find

$y=-\frac{1}{2}\left ( \frac{4}{5} \right )+2=\frac{8}{5}$

So we have the point $(\frac{4}{5},\frac{8}{5})$ , which is closest to the origin. We can now find its distance from that origin.

$d=\sqrt{\left (\frac{4}{5} \right )^2+\left ( \frac{8}{5} \right )^2}$

$d=\sqrt{\frac{16}{5}}=\frac{4}{\sqrt{5}}$

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